MKdV-Burgers方程衰减振荡解的近似解和误差估计

Approximate Damped Oscillatory Solutions of MKdV-Burgers Equation and Their Error Estimation

研究了MKdV-Burgers方程衰减振荡解近似解的求解及其误差估计问题.利用平面动力系统理论对MKdV-Burgers方程的行波解所对应的动力系统作了定性分析,给出了其在不同参数条件下的全局相图和有界行波解存在的条件和个数.研究了该方程有界行波解的波形与耗散系数之间的关系,给出了表征耗散作用大小的两个临界值,得到了当耗散系数α大于某个临界值时方程有界行波解的波形表现为单调扭状孤波、当耗散系数小于某个临界值时方程有界行波解的波形表现为衰减振荡波的结论;求得了该方程在无耗散作用情况下所有可能的3种钟状孤波解.根据衰减振荡解对应的解轨线在相图中的演化关系,并利用假设待定法,求得了该方程的衰减振荡解的近似解.最后,根据齐次化原理的思想,通过建立反映衰减振荡解精确解和近似解间关系的积分方程,得到了所求衰减振荡近似解的误差估计,其误差是以指数形式速降的无穷小量。

The theory of planar dynamical systems was utilized to qualitatively analyse the dynamical systems to which correspond the traveling wave solutions of MKdV-Burgers equation.The global phase portraits under different parameter conditions were presented and the conditions when the bounded traveling wave solutions exist.Furthermore,the relations between the wave profiles of bounded traveling wave solutions and the dissipation coefficient α in the equation were investigated.Two critical values which describe the level of dissipating action were obtained and it was found,that a bounded traveling wave appears as a kink profile solitary wave if α is greater than some critical value,while it appears as a damped oscillatory wave if α is less than some critical value.Three kinds of bell profile solitary wave solutions of MKdV-Burgers equation with no dissipation effect were obtained.Based on the above discussion and according to the evolution relations of orbits in the global phase portraits,all approximate damped oscillatory solutions were obtained by using undetermined coefficients method.In the light of the idea of homogenization principle,the integral equations reflecting the relations between exact and approximate damped oscillatory solutions were established and the error estimation of these approximate solutions was achieved.